Playground
Order
Consider a scenario similar to metal rod with
temperature varying across the length.
o
Like a metal spoon dipped in a hot curry and removed
o
Or a piece of iron heated in a furnace
and then removed
And we want to study the heat flow, how the
temperature at different points vary with time
P
Consider the met
al rod to be 1 D
Neglect
o
Heat loss to environment
o
Any heating s
ource
∙
∙
∙
The 1 D rod is thus isolated
X
∙
∙
∙
∙
Consider the metal rod to be a
cylindrical rod
o
Of radius R and length L
∙
It is given that the rod is not in
contact with any heat source
∙
We assume that no heat is lost
to surroundings
V
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
L
∙
The time rate of change of temperature of a point x is proportional to
the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
𝑄
(
𝑥
,
𝑡
)
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where c, k’ are constants.
where Q (x,t) is the heat absorbed by the
element at x
T
•
•
From calculus we have
[𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where
𝑘
=
𝑐
(
𝑑
𝑥
2
)
which describes how T at some x varies with t.
G
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
R
•
•
Place t
he rod in a mathematical grid
Let T (x, t) denote the temperature of
any mass point x at time t.
Z
∙
The time rate of change of temperature of a point x is proportional
to the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
where c is a constant.
M
∙
∙
From calculus we have
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
=
(
𝑑
𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
which describes how T at some x varies with t. Here
𝑘
=
𝑐
(
𝑑
𝑥
2
)
.
D
BLANK
Consider a scenario similar to metal rod with
temperature varying across the length.
o
Like a metal spoon dipped in a hot curry and removed
o
Or a piece of iron heated in a furnace
and then removed
And we want to study the heat flow, how the
temperature at different points vary with time
P
Consider the met
al rod to be 1 D
Neglect
o
Heat loss to environment
o
Any heating s
ource
∙
∙
∙
The 1 D rod is thus isolated
X
∙
∙
∙
∙
Consider the metal rod to be a
cylindrical rod
o
Of radius R and length L
∙
It is given that the rod is not in
contact with any heat source
∙
We assume that no heat is lost
to surroundings
V
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
L
∙
The time rate of change of temperature of a point x is proportional to
the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
𝑄
(
𝑥
,
𝑡
)
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where c, k’ are constants.
where Q (x,t) is the heat absorbed by the
element at x
T
•
•
From calculus we have
[𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where
𝑘
=
𝑐
(
𝑑
𝑥
2
)
which describes how T at some x varies with t.
G
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
R
•
•
Place t
he rod in a mathematical grid
Let T (x, t) denote the temperature of
any mass point x at time t.
Z
∙
The time rate of change of temperature of a point x is proportional
to the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
where c is a constant.
M
∙
∙
From calculus we have
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
=
(
𝑑
𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
which describes how T at some x varies with t. Here
𝑘
=
𝑐
(
𝑑
𝑥
2
)
.
D
BLANK
Consider a scenario similar to metal rod with
temperature varying across the length.
o
Like a metal spoon dipped in a hot curry and removed
o
Or a piece of iron heated in a furnace
and then removed
And we want to study the heat flow, how the
temperature at different points vary with time
P
Consider the met
al rod to be 1 D
Neglect
o
Heat loss to environment
o
Any heating s
ource
∙
∙
∙
The 1 D rod is thus isolated
X
∙
∙
∙
∙
Consider the metal rod to be a
cylindrical rod
o
Of radius R and length L
∙
It is given that the rod is not in
contact with any heat source
∙
We assume that no heat is lost
to surroundings
V
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
L
∙
The time rate of change of temperature of a point x is proportional to
the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
𝑄
(
𝑥
,
𝑡
)
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where c, k’ are constants.
where Q (x,t) is the heat absorbed by the
element at x
T
•
•
From calculus we have
[𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where
𝑘
=
𝑐
(
𝑑
𝑥
2
)
which describes how T at some x varies with t.
G
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
R
•
•
Place t
he rod in a mathematical grid
Let T (x, t) denote the temperature of
any mass point x at time t.
Z
∙
The time rate of change of temperature of a point x is proportional
to the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
where c is a constant.
M
∙
∙
From calculus we have
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
=
(
𝑑
𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
which describes how T at some x varies with t. Here
𝑘
=
𝑐
(
𝑑
𝑥
2
)
.
D
BLANK
Consider a scenario similar to metal rod with
temperature varying across the length.
o
Like a metal spoon dipped in a hot curry and removed
o
Or a piece of iron heated in a furnace
and then removed
And we want to study the heat flow, how the
temperature at different points vary with time
P
Consider the met
al rod to be 1 D
Neglect
o
Heat loss to environment
o
Any heating s
ource
∙
∙
∙
The 1 D rod is thus isolated
X
∙
∙
∙
∙
Consider the metal rod to be a
cylindrical rod
o
Of radius R and length L
∙
It is given that the rod is not in
contact with any heat source
∙
We assume that no heat is lost
to surroundings
V
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
L
∙
The time rate of change of temperature of a point x is proportional to
the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
𝑄
(
𝑥
,
𝑡
)
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where c, k’ are constants.
where Q (x,t) is the heat absorbed by the
element at x
T
•
•
From calculus we have
[𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where
𝑘
=
𝑐
(
𝑑
𝑥
2
)
which describes how T at some x varies with t.
G
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
R
•
•
Place t
he rod in a mathematical grid
Let T (x, t) denote the temperature of
any mass point x at time t.
Z
∙
The time rate of change of temperature of a point x is proportional
to the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
where c is a constant.
M
∙
∙
From calculus we have
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
=
(
𝑑
𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
which describes how T at some x varies with t. Here
𝑘
=
𝑐
(
𝑑
𝑥
2
)
.
D
BLANK
Consider a scenario similar to metal rod with
temperature varying across the length.
o
Like a metal spoon dipped in a hot curry and removed
o
Or a piece of iron heated in a furnace
and then removed
And we want to study the heat flow, how the
temperature at different points vary with time
P
Consider the met
al rod to be 1 D
Neglect
o
Heat loss to environment
o
Any heating s
ource
∙
∙
∙
The 1 D rod is thus isolated
X
∙
∙
∙
∙
Consider the metal rod to be a
cylindrical rod
o
Of radius R and length L
∙
It is given that the rod is not in
contact with any heat source
∙
We assume that no heat is lost
to surroundings
V
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
L
∙
The time rate of change of temperature of a point x is proportional to
the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
𝑄
(
𝑥
,
𝑡
)
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where c, k’ are constants.
where Q (x,t) is the heat absorbed by the
element at x
T
•
•
From calculus we have
[𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where
𝑘
=
𝑐
(
𝑑
𝑥
2
)
which describes how T at some x varies with t.
G
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
R
•
•
Place t
he rod in a mathematical grid
Let T (x, t) denote the temperature of
any mass point x at time t.
Z
∙
∙
From calculus we have
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
=
(
𝑑
𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
which describes how T at some x varies with t. Here
𝑘
=
𝑐
(
𝑑
𝑥
2
)
.
D
BLANK
∙
The time rate of change of temperature of a point x is proportional
to the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
where c is a constant.
M
∙
The time rate of change of temperature of a point x is proportional
to the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
where c is a constant.
M
Consider a scenario similar to metal rod with
temperature varying across the length.
o
Like a metal spoon dipped in a hot curry and removed
o
Or a piece of iron heated in a furnace
and then removed
And we want to study the heat flow, how the
temperature at different points vary with time
P
Consider the met
al rod to be 1 D
Neglect
o
Heat loss to environment
o
Any heating s
ource
∙
∙
∙
The 1 D rod is thus isolated
X
∙
∙
∙
∙
Consider the metal rod to be a
cylindrical rod
o
Of radius R and length L
∙
It is given that the rod is not in
contact with any heat source
∙
We assume that no heat is lost
to surroundings
V
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
L
∙
The time rate of change of temperature of a point x is proportional to
the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
𝑄
(
𝑥
,
𝑡
)
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where c, k’ are constants.
where Q (x,t) is the heat absorbed by the
element at x
T
•
•
From calculus we have
[𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where
𝑘
=
𝑐
(
𝑑
𝑥
2
)
which describes how T at some x varies with t.
G
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
R
•
•
Place t
he rod in a mathematical grid
Let T (x, t) denote the temperature of
any mass point x at time t.
Z
∙
∙
From calculus we have
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
=
(
𝑑
𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
which describes how T at some x varies with t. Here
𝑘
=
𝑐
(
𝑑
𝑥
2
)
.
D
BLANK
∙
∙
From calculus we have
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
=
(
𝑑
𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
which describes how T at some x varies with t. Here
𝑘
=
𝑐
(
𝑑
𝑥
2
)
.
D
∙
The time rate of change of temperature of a point x is proportional
to the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
where c is a constant.
M
Consider a scenario similar to metal rod with
temperature varying across the length.
o
Like a metal spoon dipped in a hot curry and removed
o
Or a piece of iron heated in a furnace
and then removed
And we want to study the heat flow, how the
temperature at different points vary with time
P
Consider the met
al rod to be 1 D
Neglect
o
Heat loss to environment
o
Any heating s
ource
∙
∙
∙
The 1 D rod is thus isolated
X
∙
∙
∙
∙
Consider the metal rod to be a
cylindrical rod
o
Of radius R and length L
∙
It is given that the rod is not in
contact with any heat source
∙
We assume that no heat is lost
to surroundings
V
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
L
∙
The time rate of change of temperature of a point x is proportional to
the difference in
temperature with the neighbouring points.
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
+
[
𝑇
(
𝑥
−
𝑑𝑥
)
−
𝑇
(
𝑥
)
]
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
∝
𝑄
(
𝑥
,
𝑡
)
Implies,
𝑑𝑇
(
𝑥
,
𝑡
)
𝑑𝑡
=
𝑐
[
𝑇
(
𝑥
+
𝑑𝑥
)
−
2
𝑇
(
𝑥
)
+
𝑇
(
𝑥
−
𝑑𝑥
)
]
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where c, k’ are constants.
where Q (x,t) is the heat absorbed by the
element at x
T
•
•
From calculus we have
[𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥
2
)
𝑑
2
𝑇
𝑑
𝑥
2
Combining both equations we get,
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑐
(
𝑑
𝑥
2
)
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
or
𝜕𝑇
(
𝑥
,
𝑡
)
𝜕𝑡
=
𝑘
𝜕
2
𝑇
𝜕
𝑥
2
+
𝑘
′
𝑄
(
𝑥
,
𝑡
)
where
𝑘
=
𝑐
(
𝑑
𝑥
2
)
which describes how T at some x varies with t.
G
∙
∙
The temperature (T) of the rod is different at different
points and vary with time
To study how T evolves, consider the rod to be made
of mass points separated by dx
•
•
•
We consider the limit dx
→
0
R
•
•
Place t
he rod in a mathematical grid
Let T (x, t) denote the temperature of
any mass point x at time t.
Z
BLANK
Finish